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An almost sure result for path lengths in binary search trees

Part of: Theory of computing Limit theorems Combinatorial probability Theory of data

Published online by Cambridge University Press:  22 February 2016

F. M. Dekking  and
L. E. Meester
F. M. Dekking*
Affiliation:
Thomas Stieltjes Institute for Mathematics and Delft University of Technology
L. E. Meester*
Affiliation:
Thomas Stieltjes Institute for Mathematics and Delft University of Technology
*
∗ Postal address: Department of ITS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.
∗ Postal address: Department of ITS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.
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Abstract

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

Keywords

Binary search treerandom permutation modelpath lengthalmost sure convergencequicksort distribution

MSC classification

Primary: 68P05: Data structures 60C05: Combinatorial probability
Secondary: 60F15: Strong theorems 68Q25: Analysis of algorithms and problem complexity
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 363 - 376
Copyright
Copyright © Applied Probability Trust 2003 

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